The linear expansivity of brass is 2 x 10$^{-5}$ C$^{-1}$. If the volume of a piece of brass is 15.00cm at 0°C, what is the volume at 100°C

1. Analyze the Question

We are given the linear expansivity ($\alpha$) of brass and its initial volume ($V_0$) at an initial temperature ($\theta_1$). We need to find the final volume ($V_2$) after the temperature increases to $\theta_2$.

Given data:

• Linear expansivity ($\alpha$) = $2 \times 10^{-5} \, ^\circ\text{C}^{-1}$
• Initial volume ($V_0$) = $15.00 \, \text{cm}^3$
• Initial temperature ($\theta_1$) = $0 \, ^\circ\text{C}$
• Final temperature ($\theta_2$) = $100 \, ^\circ\text{C}$
• Change in temperature ($\Delta\theta$) = $100 - 0 = 100 \, ^\circ\text{C}$

2. Explain the Concept

When a solid is heated, it expands in all directions. The relationship between linear expansivity ($\alpha$) and cubic (volume) expansivity ($\gamma$) is given by:

The formula for the change in volume ($\Delta V$) is:

Therefore, the final volume ($V_2$) is:

3. Show the Solution

Step 1: Calculate the cubic expansivity ($\gamma$)

Step 2: Calculate the change in volume ($\Delta V$)

Step 3: Calculate the final volume ($V_2$)

4. Identify the Answer

Comparing our result ($15.09 \, \text{cm}^3$) to the given options, it matches option B.

5. Exam Tip

Always remember the "3" in $\gamma = 3\alpha$. A common mistake students make is using the linear expansivity directly in the volume formula. Since volume is three-dimensional (length × breadth × height), you must multiply the linear expansivity by 3 to get the volume expansivity.